HISTORIA MATHEMATICA 23 (1996), 437–446ARTICLE NOS. 0041–0043

REVIEWS

Edited by CATHERINE GOLDSTEIN AND PAUL R. WOLFSON

All books, monographs, journal articles, and other publications (including films and othermultisensory materials) relating to the history of mathematics are abstracted in the AbstractsDepartment. The Reviews Department prints extended reviews of selected publications.

Materials for review, except books, should be sent to the Abstracts Editor, Professor DavidE. Zitarelli, Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122,U.S.A. Books in English for review should be sent to Professor Paul R. Wolfson, Departmentof Mathematics, West Chester University, West Chester, Pennsylvania 19383, U.S.A. Books inother languages for review should be sent to Professor Catherine Goldstein, Batiment 425Mathematiques, Universite de Paris-Sud, F-91405 Orsay Cedex, France.

Most reviews are solicited. However, colleagues wishing to review a book are invited tomake their wishes known to the appropriate Book Review Editor. (Requests to review bookswritten in the English language should be sent to Prof. Paul R. Wolfson at the above address;requests to review books written in other languages should be sent to Prof. Catherine Goldsteinat the above address.) We also welcome retrospective reviews of older books. Colleaguesinterested in writing such reviews should consult first with the appropriate Book ReviewEditor (as indicated above, according to the language in which the book is written) toavoid duplication.

Development of Mathematics, 1900–1950. Edited by Jean-Paul Pier. Basel/Boston/Berlin (Birkhauser-Verlag). 1994. 729 pp. DM 118. £ 43.

Reviewed by JEREMY GRAY

Faculty of Mathematics and Computing, Open University, Milton Keynes, MK7 6AA, England

Will we ever know the history of mathematics in the 20th century? The mostlikely answer is surely that we, at least, will not. Not because too much is too wellhidden or too difficult to recover, as is the case with the political history of thatviolent century. Not because histories that are too Eurocentric are nowadays rightlydismissed as too narrow, thus enlarging the task impossibly. But simply becausewhat is on the surface, written in a few well-known languages and published inaccessible journals and books, is too much and too hard for us to master. Learningthe history with all its imprecisions is harder, in some ways, than learning themathematics (albeit easier in others), and we can wonder if we are asking the rightquestions. Before turning to the essays this book offers, consider its first 34 pages.This is a chronological list of 1000 important papers and books published between1900 and 1950, compiled by asking some 50 experts. It cannot reasonably be sup-posed that any one person will have read these works and formed a fresh andaccurate opinion of them, let alone of their much larger penumbra. Nor, of course,

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has the editor asked just one person to write the book. Instead we have 12 peoplewho write about different aspects, and still much is left out. It could not be otherwise.Nor will the reader of this review fail to see that this single reviewer has his limita-tions.

The essays agree in a number of methodological ways. They present the historyof mathematics as the growth of numerous branches of mathematics and so as thehistory of ideas. There are no quantitative methods here, no attempt to describewhole communities, almost no social history, and very little biography. There isvery little attempt to show why the chosen subjects matter, or mattered in theirown day. Why algebraic topology gets 115 pages of text and 6 of bibliography butpartial differential equations only get 21 pages of text and 16 of bibliography is leftunexplained. But we know the answer; it simply reflects the expertise available toPier and his colleagues at that time faced with the impossible task of conveyingthe history of mathematics, 1900–1950. Even so, the choice of topics, with so muchon mathematical logic (137 pages of text, 47 of bibliography), geometry subsumedby topology, and no algebra, makes one want to ask what belongs to mathematics,and what not?

As it happens, the essay on the history of topology is a gem. Jean Dieudonnewas able to stand back from the details described in his much longer book [1]and give a remarkable account, with hints of motivation, helpful examples, andindications of personal and intellectual connections; the result is algebraic topologyas a living subject, driven by the curiosity and the insights of numerous leadingmathematicians (and not particularly constrained by the terminus of 1950 either).The omissions are part of the success of the story; the reader travels light, seeingworthwhile problems and their (sometimes) partial solutions.

Two shorter essays follow which can be read as offering methodological spice,even a hint of dissent. Doob’s essay, ‘‘The Development of Rigor in MathematicalProbability, (1900–1950),’’ is a delight. The title already indicates that a sensibleway through a vast amount of material has been found. Almost at once Doobwrites: ‘‘Specific results are mentioned only in so far as they are important in thehistory of the logical development of mathematical probability.’’ This is a goodcriterion for selection. Then he offers three famous opinions (by Planck, Poincare,and Hermite) on progress in their subjects and follows it with three on the law oflarge numbers (that it is a theorem, a proposition, and a fact) and five on thedefinition of probability. Doob modestly deduces that this conflict of opinionrequires one to separate mathematical probability from its real world applications,but it does much more: it establishes that there was a real debate about asubstantial and difficult issue. Then he turns to measure theory, notes some ofthe work on Brownian motion, work of Borel, and Kolmogorov’s memoir of1933 (which was not immediately accepted). The author concludes by criticisingthe strange way in which many mathematicians try to hold measure theory andprobability theory apart.

The next short essay, by Fichera, focuses on the evaluation of Volterra’s workon functional analysis. He draws out its presuppositions and shows accordingly

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what it could do and where it was misleading, thus explaining its impact—ultimatelynegative—in Italy, because of Volterra’s prestige. This vindicates the well-placedcriticism of Dieudonne’s account of the same material with which Fichera beginshis essay: the distinction between doing history and writing an historical review.The latter exercise admits modern insights denied to the protagonists and inviteshistorical misrepresentation. Reader—beware.

The other long essay, by Guillaume on mathematical logic, raises an awkwardhistorical problem. No one disputes the profundity of the topic, but the field isdeep and narrow, its relation to the rest of mathematics is difficult to elucidate andsince the 1930s has become tenuous. Guillaume begins by invoking a number ofthe 19th-century sources of mathematical logic and then surveys a wide variety ofissues in set theory, logic, and the foundations of mathematics. Despite the clarityof each paragraph, the overall effect is confusing. One is left wondering what theimport of all this material was (in its day) and is (for the historian). In 137 pagesof text and 47 bibliography, there has been some attempt at completeness, eventhough the author rightly admits at the very start that a completely faithful historywould require several hundred pages. What, indeed, was the historical question atissue here? Most likely an attempt to say a bit about almost everything that seemsto have lasted. Now it is a commonplace that Godel’s theorems put an end toHilbert’s programme to rigorise mathematics and with it the serious commitmentof mathematicians to mathematical logic and the foundations of mathematics. Thefamous indifference of Bourbaki to these issues belongs here. But a commonplaceneed not be true. One might reasonably ask an historian to confront the question,and one might well ask Guillaume because his essay takes a sharp turn with Godel’swork. After the mid-1930s it becomes an account of specialists doing difficult techni-cal work, innocently reinforcing our sense (if we are not logicians) of its remoteness.But although Godel’s theorems are said to have been an earthquake—and ninespecific issues are listed as flowing from them as well as quite an amount of literature(good and not so good)—one misses any sense of what their historical significancewas. If indeed the 19th-century programmes to make sense of mathematics allperished at this moment, then how and why? (I owe to a conversation with RayMonk the realisation that the matter really is not simple.) Is the subsequent workas a result as dry, even irrelevant, as it seems? Better use of the growing historicalliterature might have helped to shape this essay.

The reader now enters the second half of the book: eight essays in just under250 pages. There is a skill needed here by all of us, for the 30-page essay is whatbooks and journals like. It helps if you have a topic of about the right size. Houzel’saccount of the prehistory of the Weil conjectures is one such topic. A small numberof mathematicians progressively elucidated an area until one could formulate aseries of rich questions which even as they were asked issued a profound challengeto the existing techniques (a rich mixture of algebraic number theory and algebraicgeometry). Kahane’s essay on Taylor series and Brownian motion addresses theissue of what mathematicians meant when they said something held in general; thequestion is a good one and the answer instructive. Mahwin’s account of how ques-

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tions in nonlinear ordinary differential equations led to the work of, amongstothers, Leray and Schauder is rapid and dense, but it shares with Dieudonne’s thefeel that questions and problems led, in a natural way, to new methods and newproblems. One picks up a sense of excitement.

The editor of the book, Jean-Paul Pier, contributed an essay on integration andmeasure (but not probability). It shares with Mahwin’s the ability to connect things,to show this mathematician responding to that one. The emphasis is on the powerof the techniques being developed, and it would have been interesting to see themdo more; they were not, after all, put forward just for their own sake. The connectionwith functional analysis is left until the appearance of Schwartz, and that is also apity, but this is the inevitable downside of a correct decision to pursue one aspectand make it intelligible.

The essays by Lichnerowicz, Nirenberg, Hayman, and Schwarz are less successful.The first of these is perhaps too short, with the result that the mathematics is toofar away to be seen clearly, and one gets generalisations where precision was needed.Nirenberg’s essay on partial differential equations is little more than a list of results(who first proved what, when). A whole book needs to be written on the topic, butin a limited space it would surely have been better to tell much less, with morespirit. Much the same can be said of Hayman’s report on topics in complex analysis,which also says little new, but with less excuse. The paradox here is that while inits day Nevanlinna’s theory was highly praised (by Hilbert and Weyl, no less), thetopic of single variable complex function theory has ever since dwindled in esteem.This is not the type of historical development this genre of book can deal with easily.Schwarz’s essay on the prime number theorem is equally factual, chronological, andunexciting.

The presence of Jean Dieudonne dominates the book. His is the first photographto appear, and the first essay, but influence is felt in less visible ways: not perhapsin the choice of topics, but in the approach to history. This book is written in thedominant mode of history of mathematics, which emphasises the mathematicalresults. I have no criticisms of that mode, provided that one admits others elsewhere.What turns out to be curiously intangible is how the same approach can produceexcitement in one essay and boredom in another. It may be partly what the reader(or reviewer) brings to the topic. It may be a literary skill. But sometimes thenarrative mode is gripping: you want to read on, to know what happened next andwhy. Sometimes the result is facts, and one longs to bring them to life with questions.The story-telling skill of Dieudonne, the astute criticism of Fichera, and the dexterityof Doob are good examples for the historian to ponder, not least because they donot point in the same direction.

REFERENCES1. J. Dieudonne, A History of Algebraic and Differential Topology, 1900–1960, Boston/Basel: Birk-

hauser, 1989.